Three--parameter weighted Hardy type inequalities (Q958844)

The authors find necessary and sufficient conditions for the validity of the following inequality: \[ \left( \int_{a}^{b} u(x) \left( \int_{a}^{x} |g(x)- g(t)|^{r} w(t)\,dt \right)^{\frac{q}{r}} \,dx \right)^{\frac{1}{q}} \leq C \left( \int_{a}^{b} v(x)|g'(x)|^{p}\,dx \right)^{\frac{1}{p}} \] where \ \(u(\cdot), v(\cdot), w(\cdot)\) are weight functions, \(0<r< \infty\), \(1\leq p \leq q < \infty\) and \(C\) is the best constant.